On the Plants Leaves Boundary, "Jupe à Godets" and Conformal Embeddings

Abstract

The stable profile of the boundary of a plant's leaf fluctuating in the direction transversal to the leaf's surface is described in the framework of a model called a "surface à godets". It is shown that the information on the profile is encoded in the Jacobian of a conformal mapping (the coefficient of deformation) corresponding to an isometric embedding of a uniform Cayley tree into the 3D Euclidean space. The geometric characteristics of the leaf's boundary (like the perimeter and the height) are calculated. In addition a symbolic language allowing to investigate statistical properties of a "surface à godets" with annealed random defects of curvature of density $q$ is developed. It is found that at $q=1$ the surface exhibits a phase transition with critical exponent $\alpha =1/2$ from the exponentially growing to the flat structure.

17 pages (revtex), 8 eps-figures, to appear in Journal of Physics A

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